Crack in a material-bond lattice

A square-cell lattice is considered consisting of point masses at its knots connected by linearly elastic bonds of nonzero density. Steady-state crack propagation is studied. A general relation between the knot mass and the bond mass is assumed; however, a detailed analytical examination is made for the material-bond lattice with no concentrated masses. It is assumed that the crack divides the bond in half, and the broken bonds remain in the lattice structure. In this model, the fracture energy of the bond is ignored, and hence the local fracture energy of the lattice is zero. The classical formulation in terms of critical stresses is accepted. The macrolevel energy release does exist. The macrolevel energy release rate as a function of the crack speed is found and compared with that for the massless-bond lattice of the same averaged density. While in the main, the dependencies for these two models are similar, there are some essential differences. For the lattice with no concentrated masses this function appears discontinuous. There exists a region where the crack speed is insensitive to the variation of the macrolevel energy release rate. The admissible regions of the crack speeds for the considered two lattice models differ greatly. For the massless-bond lattice this region is rather wide, while for the other it is very narrow. Mathematically, it is of interest that some details of the factorization depend on whether the ratio of the crack speed to the wave speed is rational and, if so, whether it can be represented as a ratio of two odd numbers.